<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">% è¡¨é¡Œ  å†…å¤–å¢ƒç•Œçƒé¢ã®å›žè»¢ã«ã‚ˆã‚Šå¼•ãèµ·ã“ã•ã‚Œã‚‹çƒæ®»å†…ã®ç²˜æ€§æµä½“ã®æµã‚Œ
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\Dtitle[å†…å¤–å¢ƒç•Œçƒé¢ã®å›žè»¢ã«ã‚ˆã‚‹çƒæ®»å†…ã®ç²˜æ€§æµä½“]
       {å†…å¤–å¢ƒç•Œçƒé¢ã®å›žè»¢ã«ã‚ˆã‚Šå¼•ãèµ·ã“ã•ã‚Œã‚‹çƒæ®»å†…ã®ç²˜æ€§æµä½“ã®æµã‚Œ}
\Dauthor{ç«¹åºƒ çœŸä¸€}
\Ddate{å¹³æˆ 17 å¹´ 3 æœˆ 1 æ—¥} 
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\begin{document}

\maketitle

\section{ã¯ã˜ã‚ã«}

  ã“ã“ã§ã¯ä¸­å¿ƒã‚’å…±æœ‰ã™ã‚‹ãŒç•°ãªã‚‹å›žè»¢è§’é€Ÿåº¦ã§å›žè»¢ã™ã‚‹ 2 ã¤ã®çƒé¢(çƒæ®»)ã«
  æŒŸã¾ã‚ŒãŸé&nbsp;˜åŸŸå†…ã®éžåœ§ç¸®ç²˜æ€§æµä½“ã®é‹å‹•ã‚’è€ƒãˆã‚‹. 

\section{æ”¯é…æ–¹ç¨‹å¼}


\section{è§£æžè§£}

  \subsection{æ…£æ€§é&nbsp;…ãŒç„¡è¦–ã§ãã‚‹å&nbsp;´åˆ(ã‚¹ãƒˆãƒ¼ã‚¯ã‚¹è¿‘ä¼¼)ã®è§£}
  
    å†…å¤–çƒæ®»ã®å›žè»¢ã«ã‚ˆã‚Šå¼•ãèµ·ã“ã•ã‚Œã‚‹çƒæ®»ä¸­ã®æµã‚Œã«ã¤ã„ã¦, 
    å›žè»¢è§’é€Ÿåº¦ãŒååˆ†ã«å°ã•ãç²˜æ€§é&nbsp;…ã«æ¯”ã¹ã¦æ…£æ€§é&nbsp;…ãŒç„¡è¦–ã§ãã‚‹
    çŠ¶æ³ã§ã®è§£ã‚’è¦‹å‡ºã™\footnotemark[6]
    ($Re=\Omega_i D^2/\nu, \Omega_o D^2/\nu \ll 1)$. 
    
    \footnotetext[6]
       {Landau and Lifshitz, Vol.6, Fluid Mechanics (1987), Section 20, 
        Problem 1. ã‚’å‚ç…§ã®ã“ã¨. }
    
    å†…å¤–çƒæ®»ãŒãã‚Œãžã‚Œä¸€å®šã®è§’é€Ÿåº¦ $\Dvect{\Omega_i},\Dvect{\Omega_o}$
    ã§å›žè»¢ã—ã¦ã„ã‚‹ã¨ã™ã‚‹. 
    æº€ãŸã™ã¹ãæ–¹ç¨‹å¼ã¯, å¤–åŠ›ãŒãªãæ…£æ€§é&nbsp;…ã‚’ç„¡è¦–ã—ãŸå¼, 
    %
    \begin{displaymath}
       L_2\Dlapla \psi = 0,   L_2\Dlapla\Dlapla \phi =0, 
    \end{displaymath}
    %
    å¢ƒç•Œã§ã®æ¡ä»¶ã¯ç²˜ç€æ¡ä»¶ã§ã‚ã‚‹. 
    %
    \begin{eqnarray*}
      &amp; &amp; \phi = \DP{\phi}{r} = 0, \quad
          \psi = (\Dvect{r}\cdot\Dvect{\Omega_i})
          \quad \mbox{at} \quad r=r_i=\frac{\zeta}{1-\zeta}, \\
      &amp; &amp; \phi = \DP{\phi}{r} = 0, \quad
          \psi = (\Dvect{r}\cdot\Dvect{\Omega_o})
          \quad \mbox{at} \quad r=r_o=\frac{1}{1-\zeta}. 
    \end{eqnarray*}
    %
    ãã“ã§
    \begin{displaymath}
      \phi=0, \quad
      \psi= A(r)(\Dvect{r}\cdot\Dvect{\Omega_i})
            +B(r)(\Dvect{r}\cdot\Dvect{\Omega_o})
    \end{displaymath}
    %
    ã®å½¢ã§è§£ã‚’æ±‚ã‚ã‚‹. ã“ã“ã§ $A(r),B(r)$ ã¯å¢ƒç•Œæ¡ä»¶
    %
    \begin{displaymath}
       A(r=r_i) = 1, \quad  A(r=r_o) = 0, \quad
      B(r=r_i) = 0, \quad   B(r=r_o) = 1,
    \end{displaymath}
    %
    ã‚’æº€ãŸã™é–¢æ•°ã§ã‚ã‚‹. 
    %
    ä»®å®šã—ãŸè§£ã®å½¢ã‚’æ”¯é…æ–¹ç¨‹å¼ã«ä»£å…¥ã™ã‚‹ã¨, $A(r), B(r)$ ã®
    æº€ãŸã™ã¹ãæ–¹ç¨‹å¼ãŒ\footnotemark[9]
    \footnotetext[9]{
      ç¬¬ 1 é&nbsp;…ç›®ã&nbsp;ã‘ä»£å…¥ã™ã‚‹ã¨
      \begin{eqnarray*}
        &amp; &amp; L_2\Dlapla A(r)(\Dvect{r}\cdot\Dvect{\Omega_i})
          = L_2\left(\frac{1}{r^2}\DP{}{r}r^2\DP{}{r}- \frac{L_2}{r^2}\right)
              A(r)(\Dvect{r}\cdot\Dvect{\Omega_i})\\
        &amp;=&amp; L_2\frac{1}{r^2}\DP{}{r}
              \left(r^2\DP{}{r}[A(r)(\Dvect{r}\cdot\Dvect{\Omega_i})]\right)
           - \frac{L_2 L_2 A(r)(\Dvect{r}\cdot\Dvect{\Omega_i})}{r^2}\\
        &amp;=&amp; \frac{1}{r^2}\DP{}{r}
            \left(r^2\DP{}{r}[A(r)L_2(\Dvect{r}\cdot\Dvect{\Omega_i})]\right)
           - \frac{A(r)}{r^2}L_2 L_2(\Dvect{r}\cdot\Dvect{\Omega_i})\\
        &amp;=&amp; \frac{1}{r^2}\DP{}{r}
            \left(r^2\DP{}{r}[A(r)2(\Dvect{r}\cdot\Dvect{\Omega_i})]\right)
           - \frac{A(r)}{r^2}4(\Dvect{r}\cdot\Dvect{\Omega_i})\\
        &amp;=&amp; \frac{2}{r^2}\DP{}{r}
            \left(r^2\DP{}{r}[A(r)(\Dvect{r}\cdot\Dvect{\Omega_i})]\right)
           - \frac{4A(r)(\Dvect{r}\cdot\Dvect{\Omega_i})}{r^2}\\
        &amp;=&amp; \frac{2}{r^2}\DP{}{r}
            \left(r^2(\Dvect{r}\cdot\Dvect{\Omega_i})\DP{A(r)}{r}
            + r^2A(r)\frac{(\Dvect{r}\cdot\Dvect{\Omega_i})}{r}\right)
           - \frac{4A(r)(\Dvect{r}\cdot\Dvect{\Omega_i})}{r^2}\\
        &amp;=&amp; \frac{2}{r^2}\DP{}{r}
              \left(r^2(\Dvect{r}\cdot\Dvect{\Omega_i})\DP{A(r)}{r}\right)
           +\frac{2}{r^2}\DP{}{r}
              [rA(r)(\Dvect{r}\cdot\Dvect{\Omega_i})]
           - \frac{4A(r)(\Dvect{r}\cdot\Dvect{\Omega_i})}{r^2}\\
        &amp;=&amp; \frac{2(\Dvect{r}\cdot\Dvect{\Omega_i})}{r^2}\DP{}{r}
              \left(r^2\DP{A(r)}{r}\right)
            +2\DP{A(r)}{r}
               \frac{(\Dvect{r}\cdot\Dvect{\Omega_i})}{r}
            +\frac{2(\Dvect{r}\cdot\Dvect{\Omega_i})}{r^2}
              \DP{}{r}[rA(r)]
            +2\frac{A(r)}{r}\frac{(\Dvect{r}\cdot\Dvect{\Omega_i})}{r}\\
          &amp;&amp;- \frac{4A(r)(\Dvect{r}\cdot\Dvect{\Omega_i})}{r^2}\\
        &amp;=&amp; 2(\Dvect{r}\cdot\Dvect{\Omega_i})\DP[2]{A(r)}{r}
           + \frac{4(\Dvect{r}\cdot\Dvect{\Omega_i})}{r}\DP{A(r)}{r}
           +2\DP{A(r)}{r}\frac{(\Dvect{r}\cdot\Dvect{\Omega_i})}{r}
           +\frac{2(\Dvect{r}\cdot\Dvect{\Omega_i})}{r}
              \DP{A(r)}{r}
           +\frac{2(\Dvect{r}\cdot\Dvect{\Omega_i})A(r)}{r^2}\\
          &amp;&amp;- \frac{2A(r)(\Dvect{r}\cdot\Dvect{\Omega_i})}{r^2}\\
        &amp;=&amp; 2(\Dvect{r}\cdot\Dvect{\Omega_i})\DP[2]{A(r)}{r}
           + \frac{8(\Dvect{r}\cdot\Dvect{\Omega_i})}{r}\DP{A(r)}{r}\\
        &amp;=&amp; 2(\Dvect{r}\cdot\Dvect{\Omega_i})
              \left(\DP[2]{A(r)}{r}+\frac{4}{r}\DP{A(r)}{r}\right)
      \end{eqnarray*}
      ç¬¬ 2 é&nbsp;…ç›®ã«ã¤ã„ã¦ã‚‚åŒæ§˜ã«è¨ˆç®—ã§ãã‚‹. 
      $\Dvect{\Omega_i}$ ã¨ $\Dvect{\Omega_o}$ ãŒç•°ãªã‚‹å&nbsp;´åˆã«é™ã‚Š
      $(\Dvect{r}\cdot\Dvect{\Omega_i})$ ã¨
      $(\Dvect{r}\cdot\Dvect{\Omega_o})$ ãŒæ’ç­‰çš„ã« 0 ã¨ãªã‚‰ãªã„ã®ã§
      ãã‚Œã‚‰ã®ä¿‚æ•°ã‹ã‚‰ $A, B$ ã®å¾®åˆ†æ–¹ç¨‹å¼ãŒæ±‚ã‚ã‚‰ã‚Œã‚‹. 
    }
    
    \begin{displaymath}
       \DD[2]{A(r)}{r}+\frac{4}{r}\DD{A(r)}{r}=0, \quad
       \DD[2]{B(r)}{r}+\frac{4}{r}\DD{B(r)}{r}=0. 
    \end{displaymath}
    %
    $A(r),B(r)=r^n$ ã®å½¢ã‚’ä»®å®šã™ã‚‹ã¨, 
    \begin{displaymath}
      n(n-1)r^{n-2} + 4n r^{n-2} =0, \quad n(n+3)=0, 
      \quad \mbox{i.e.} \quad n=0,-3
    \end{displaymath}
    %
    ã—ãŸãŒã£ã¦, 
    \begin{displaymath}
      A(r) = C_1 + \frac{C_2}{r^3}, \quad
      B(r) = C_3 + \frac{C_4}{r^3}
    \end{displaymath}
    ã¨ãªã‚‹. ã“ã“ã§ $C_1,C_2,C_3,C_4$ ã¯å®šæ•°ã§ã‚ã‚‹. 
    %
    å¢ƒç•Œæ¡ä»¶ã‚ˆã‚Š
    \begin{displaymath}
        C_1 + \frac{C_2}{r_i^3}=1, \quad
        C_1 + \frac{C_2}{r_o^3}=0, \quad
        C_3 + \frac{C_4}{r_i^3}=0, \quad
        C_3 + \frac{C_4}{r_o^3}=1,
    \end{displaymath}
    %
    \begin{eqnarray*}
      &amp; &amp; C_1 = \frac{1}{1-r_o^3/r_i^3}= -\frac{r_i^3}{r_o^3-r_i^3}, \quad
          C_2 = \frac{r_i^3r_o^3}{r_o^3-r_i^3}, \\
      &amp; &amp; C_3 = \frac{1}{(1-r_i^3/r_o^3)}= \frac{r_o^3}{r_o^3-r_i^3}, \quad
          C_4 = -\frac{r_i^3r_o^3}{r_o^3-r_i^3}.
    \end{eqnarray*}
    %
    ã‚ˆã£ã¦
    \begin{eqnarray*}
      \phi&amp;=&amp;0, \\
      \psi&amp;=&amp; \left( -\frac{r_i^3}{r_o^3-r_i^3}
                     +\frac{r_i^3r_o^3}{r_o^3-r_i^3}\frac{1}{r^3}\right)
                 (\Dvect{r}\cdot\Dvect{\Omega_i})
            +\left( \frac{r_o^3}{r_o^3-r_i^3}
                   -\frac{r_i^3r_o^3}{r_o^3-r_i^3}\frac{1}{r^3}\right)
                 (\Dvect{r}\cdot\Dvect{\Omega_o})\\
         &amp;=&amp; -\frac{r_i^3}{r_o^3-r_i^3}\left(1-\frac{r_o^3}{r^3}\right)
                (\Dvect{r}\cdot\Dvect{\Omega_i})
            +\frac{r_o^3}{r_o^3-r_i^3}\left(1-\frac{r_i^3}{r^3}\right)
                 (\Dvect{r}\cdot\Dvect{\Omega_o}).
    \end{eqnarray*}
    %
    ç„¡æ¬¡å…ƒé‡ã§ $r_i,r_o$ ã‚’æ›¸ã‘ã°
    \begin{eqnarray*}
      \phi&amp;=&amp;0, \\
      \psi&amp;=&amp; -\frac{\zeta^3}{1-\zeta^3}
             \left(1-\frac{1}{(1-\zeta)^3 r^3}\right)
                 (\Dvect{r}\cdot\Dvect{\Omega_i})
            +\frac{1}{1-\zeta^3}
               \left(1-\frac{\zeta^3}{(1-\zeta)^3 r^3}\right)
                 (\Dvect{r}\cdot\Dvect{\Omega_o})
    \end{eqnarray*}
    %
    ç‰¹ã«å†…çƒãŒå­˜åœ¨ã—ãªã„å&nbsp;´åˆã«ã¯ $r_i\longrightarrow 0$ ã¨ã—ã¦, 
    %
    \begin{displaymath}
      \phi=0, \quad  \psi=(\Dvect{r}\cdot\Dvect{\Omega_o}).
    \end{displaymath}
    %
    çƒå†…å…¨ä½“ãŒå‰›ä½“å›žè»¢ã®è§£ã¨ãªã‚‹. 
    %
    é€†ã«å¤–å´ã®çƒãŒå­˜åœ¨ã—ãªã„æ¥µé™ã§ã¯ 
    $\Dvect{\Omega_o}=0, r_o\longrightarrow\infty$ ã¨ã—ã¦
    %
    \begin{displaymath}
      \phi = 0, \quad
      \psi = \frac{1}{r^3}(\Dvect{r}\cdot\Dvect{\Omega_i}).
    \end{displaymath}
    
    é€Ÿåº¦å&nbsp;´ã¯ãƒˆãƒ­ã‚¤ãƒ€ãƒ«ãƒãƒ†ãƒ³ã‚·ãƒ£ãƒ«ã®å®šç¾©\Deqref{é€Ÿåº¦åˆ†è§£}ã‚ˆã‚Š
    %
    \begin{displaymath}
      \Dvect{u}=\Drot(\psi\Dvect{r})
               =\Dgrad\psi\times\Dvect{r} +\psi\Drot\Dvect{r}
               =\Dgrad\psi\times\Dvect{r}.
    \end{displaymath}
    %
    ã“ã“ã§ 
    \begin{eqnarray*}
      \Dgrad [A(r)(\Dvect{r}\cdot\Dvect{\Omega_i})]\times\Dvect{r}
      &amp;=&amp; (\Dvect{r}\cdot\Dvect{\Omega_i})[\Dgrad A(r)] \times\Dvect{r}
           A(r)[\Dgrad (\Dvect{r}\cdot\Dvect{\Omega_i})]\times\Dvect{r}\\
      &amp;=&amp; A(r)\Dvect{\Omega_i}\times\Dvect{r}
    \end{eqnarray*}
    %
    $A(r)$ ã¯ $r$ ã&nbsp;ã‘ã®é–¢æ•°ãªã®ã§ $\Dgrad A(r)$ ã¯ $\Dvect{r}$ ã¨
    ä¸¦è¡Œã§ã‚ã‚Š, $[\Dgrad A(r)] \times\Dvect{r}=0 $ ã§ã‚ã‚‹ã“ã¨ã«
    æ³¨æ„ã•ã‚ŒãŸã„. 
    ã—ãŸãŒã£ã¦, 
    \begin{displaymath}
      \Dvect{u}
          = -\frac{r_i^3}{r_o^3-r_i^3}\left(1-\frac{r_o^3}{r^3}\right)
                (\Dvect{\Omega_i}\times\Dvect{r})
            +\frac{r_o^3}{r_o^3-r_i^3}\left(1-\frac{r_i^3}{r^3}\right)
                 (\Dvect{\Omega_o}\times\Dvect{r}).
    \end{displaymath}


  \subsection{ã»ã¼å‰›ä½“å›žè»¢ã«è¿‘ã„å&nbsp;´åˆã®æµã‚Œ}

    Dormy {\it et al.} (1998), Proudman (1956), Stewartson (1966) ã‚’
    å‚ç…§ã®ã“ã¨. 


\begin{Dreference}
  \item Landau, L. D., Lifshitz, E. M., 1987 : 
         Fluid Mechanics (Volume 6 of Course of theoretical physics), 
         2nd edition. Pergamon press, 539pp.

  \item Dormy, E., Cardin, P., Jault, D., 1998 : 
    MHD flow in a slightly differntially rotating spherical shell, 
    with conducting inner core, in a dipolar magnetic field. 
    {\it Earth Planet. Sci. Lett.}, {\bf 160}, 15--30. 

  \item Proudman, I., 1956 : 
    The almost-rigid rotation of viscous fluid between concentric spheres. 
    {\it J. Fluid Mech.}, {\bf 1}, 505--516. 

  \item Stewartson, K., 1966 : On almost rigid rotations. Part 2.
    {\it J. Fluid Mech.}, {\bf 26}, 131--144. 
\end{Dreference}

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